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  • Need solution for RD Sharma maths class 12 chapter 22 Algebra of Vectors exercise Multiple choice question 3

Need solution for RD Sharma maths class 12 chapter 22 Algebra of Vectors exercise Multiple choice question 3

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Answer: 3AC = 5CB

Hint: O N \perp A B

Given: Forces 3 \overrightarrow{O A}, 5 \overrightarrow{O B}  act along OA and OB. If resultant pass through C on AB then

Solution: Draw ON the perpendicular to line AB

               Let i be unit actor along ON

The Resultant Force  \vec{R}=3 \overrightarrow{O A}+5 \overrightarrow{O B}                                                ……….. (1)

The angle between i and force         

        \vec{R}, 3 \overrightarrow{O A}, 5 \overrightarrow{O B} \text { are } \angle C O N, \angle A O N, \angle B O N      

         \vec{R} \vec{i}=3 \overrightarrow{O A} \vec{i}+5 \overrightarrow{O B} \vec{i}

        \begin{aligned} &\text { R.1.cos } \angle C O N=3 \\\\ &\overrightarrow{O A} \cdot 1 . \cos \angle B O N \\\\ &R \cdot \frac{O N}{O C}=3 \overrightarrow{O A} \times \frac{O N}{O A}+5 \overrightarrow{O B} \times \frac{O N}{O B} \\\\ &\frac{R}{O C}=3+5, R=8 \end{aligned}

\overrightarrow{O C}  we know that

        \begin{aligned} &\overrightarrow{O A}=\overrightarrow{O C}+\overrightarrow{C A} \\\\ &3 \overrightarrow{O A}=3 \overrightarrow{O C}+3 \overrightarrow{C A} \end{aligned}        ..................(i)

        \begin{aligned} &\overrightarrow{O B}=\overrightarrow{O C}+\overrightarrow{C B} \\\\ &5 \overrightarrow{O B}=5 \overrightarrow{O C}+5 \overrightarrow{C B} \end{aligned}        ..................(ii)

  On adding (i) and (ii)

        \begin{aligned} &3 \overrightarrow{O A}+5 \overrightarrow{O B}=8 \overrightarrow{O C}+3 \overrightarrow{C A}+5 \overrightarrow{C B} \\ \end{aligned}

        R=8 \overrightarrow{O C}+3 \overrightarrow{C A}+5 \overrightarrow{C B} \\

        8 \overrightarrow{O C}=8 \overrightarrow{O C}+3 \overrightarrow{C A}+5 \overrightarrow{C B} \\

        |3 \overrightarrow{A C}|=|5 \overrightarrow{C B}| \\

        3 A C=5 C B

  

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